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Combining decimals
Negative decimals
You have already learned about negative integers and operations related to them. Decimals can also be positive or negative.
In the example, you could say that the \latex{ 1.2 \;m } deep hole's height is \latex{ –1.2\; m }.
The position of \latex{-1.2} on the number line:
different signs
They are located at equal distances from zero; therefore,
they are the additive inverses of each other.
\latex{-2}
\latex{-1}
\latex{0}
\latex{1}
\latex{2}
\latex{-1.2}
\latex{+1.2}
The additive inverse of \latex{(-1.2) \text{ is } -(-1.2) = +1.2}.
The additive inverse of \latex{(+1.2) \text{ is } -(+1.2) =-1.2}.
The additive inverse of \latex{(+1.2) \text{ is } -(+1.2) =-1.2}.
The absolute value of a decimal is its distance from zero on the number line:
\latex{\mid-1.2\mid= 1.2 \text{ and }\mid+1.2\mid= 1.2.}
Out of two negative decimals, the number with the smaller absolute value is the larger one.
\latex{-3}
\latex{-2}
\latex{-1}
\latex{0}
\latex{1}
\latex{-2.4}
\latex{-1.2}
\latex{-0.5}
\latex{-2.4\lt-1.2\lt-0.5}
It is valid for decimals too, that
- the sum of a number and its additive inverse is zero:
\latex{(-1.2) + (+1.2) = -1.2 + 1.2 = 0;}
- if two numbers are equal, their difference is zero:
\latex{(-1.2)-(-1.2) = 0.}
Combining decimals
When adding and subtracting decimals, do the same as in the case of integers:
- express subtractions as additions;
- eliminate the plus sign and the brackets;
- perform the operations.
For example:
- \latex{(+5.1) + (+3.04) = 5.1 + 3.04 = 8.14}
\latex{(+5.1)-(-3.04) = (+5.1) + (+3.04) = 5.1 + 3.04 = 8.14}
Since positive numbers are combined, the result is also positive.
- \latex{(+5.1) + (-3.04) = 5.1-3.04 = 2.06}
\latex{(+5.1)-(+3.04) = (+5.1) + (-3.04) = 5.1-3.04 = 2.06}
The number with the greater absolute value is positive, so the result is also positive.
- \latex{(-5.1) + (+3.04) = -5.1 + 3.04 = -2.06}
\latex{(-5.1)-(-3.04) = (-5.1) + (+3.04) = -5.1 + 3.04 = -2.06}
The number with the greater absolute value is negative, so the result is also negative.
ones
tenths
hundredths
\latex{5.}
\latex{1}
\latex{0}
\latex{3.}
\latex{8.}
\latex{1}
\latex{4}
\latex{4}
\latex{+}
\latex{-}
\latex{5.}
\latex{1}
\latex{3.}
\latex{0}
\latex{4}
\latex{2.}
\latex{0}
\latex{6}
\latex{\textcolor{gray}{0}}
\latex{\textcolor{gray}{0}}
- \latex{(-5.1) + (-3.04) = -5.1-3.04 = -8.14}
\latex{(-5.1)-(+3.04) = (-5.1) + (-3.04) = -5.1-3.04 = -8.14}
When negative numbers are combined, the result is also negative.
When you perform operations using the column method, make sure that the digits are in the correct position. The sign of the sum or the difference is determined by the absolute values of the terms and their signs.
Combining decimals in the case of several terms
Combining decimals when there are several terms is similar to the method used with integers.
- \latex{(-8.7) + (+72)-(+0.095)-(-108.03) =\\ = (-8.7) + (+72) + (-0.095) + (+108.03) =\\ = -8.7+ 72-0.095+ 108.03=\\= 72 + 108.03-8.7-0.095 =\\ = 180.03-8.795 = 171.235}
The absolute value of the sum of positive terms is greater; therefore, the result is also positive.
- \latex{(+13.5)-(-4.16) + (-234.007)=\\=(+13.5)+(+4.16) + (-234.007) =\\=13.5+4.1 -234.007 =\\ =17.66-234.007 = -216.347}
The absolute value of the negative term is greater than that of the sum of the absolute values of the positive terms; therefore, the result is negative.
\latex{7}
\latex{2.}
\latex{+}
\latex{1}
\latex{0}
\latex{8.}
\latex{0}
\latex{3}
\latex{1}
\latex{8}
\latex{0.}
\latex{0}
\latex{3}
\latex{8.}
\latex{7}
\latex{0.}
\latex{0}
\latex{9}
\latex{5}
\latex{7}
\latex{+}
\latex{1}
\latex{8}
\latex{0.}
\latex{0}
\latex{3}
\latex{-}
\latex{8.}
\latex{7}
\latex{9}
\latex{5}
\latex{5}
\latex{3}
\latex{2}
\latex{2}
\latex{1.}
\latex{7}
\latex{1}
\latex{1}
\latex{3.}
\latex{5}
\latex{4.}
\latex{1}
\latex{6}
\latex{6}
\latex{6}
\latex{7.}
\latex{1}
\latex{+}
\latex{-}
\latex{2}
\latex{3}
\latex{4.}
\latex{0}
\latex{0}
\latex{7}
\latex{1}
\latex{7.}
\latex{6}
\latex{6}
\latex{2}
\latex{1}
\latex{6.}
\latex{3}
\latex{4}
\latex{7}
\latex{8.}
\latex{9}
\latex{5}
\latex{\textcolor{gray}{0}}
\latex{\textcolor{gray}{0}}
\latex{\textcolor{gray}{0}}
\latex{\textcolor{gray}{0}}
\latex{\textcolor{gray}{0}}
\latex{\textcolor{gray}{0}}
\latex{\textcolor{gray}{0}}
You can combine the positive and negative terms separately and then combine the resulting two terms, too.
\latex{ -234.007 }
\latex{ 17.66 }
Exercises
{{exercise_number}}. Estimate the following sums and differences, rounded to the nearest ones. Perform the operations using the column method.
- \latex{0.29 + 3.708 + 0.9}
- \latex{8.63 + 21.07 + 24.8}
- \latex{17.8 + 1,100.011 + 430.97}
- \latex{17.8-10.85}
- \latex{40.067-7.09}
- \latex{2,037.9-29.056}
{{exercise_number}}. At the market, you buy \latex{ 2.5\; kg } of potatoes, one and a half \latex{ kg } of peas, \latex{ 500\; g } of strawberries, and \latex{ 0.8\; kg } of carrots. Can you carry the groceries home in a bag that can hold up to \latex{ 5\; kg } of products?
{{exercise_number}}. Fill in the missing numbers.
\latex{+1.5}
\latex{-}
\latex{-}
\latex{+}
\latex{+}
\latex{+}
\latex{+}
\latex{-}
\latex{-}
\latex{+98.7}
\latex{+4.6}
\latex{-3.5}
\latex{-24.75}
\latex{-7.7}
\latex{0}
\latex{-1.23}
a)
c)
b)
d)
{{exercise_number}}. Perform the following additions and subtractions. Check the results.
- \latex{9,713.07-405.8}
- \latex{119.7-11.97}
- \latex{6,704-582.15}
- \latex{3.7 + 28.6}
- \latex{1,629.48 + 72.059}
- \latex{7,493.025 + 6.975}
{{exercise_number}}. Combine the following numbers.
- \latex{617.54-48.14- 203.705}
- \latex{58.01-0.921-37}
- \latex{158.42-16.517- 370}
- \latex{2-0.28 + 47.044}
{{exercise_number}}. Which one is greater? By how much?
- \latex{(-7.5) + (-2.5)\quad \text{or} \quad (-7.5) -(-2.5)}
- \latex{(+103.2)-(+9.25) \text{\quad or \quad } (-103.2) + (-9.25)}
- \latex{(+4.27) + (-72.5) \text{\quad or \quad } (-72.5) + (-4.27)}
- \latex{(+1)-(+0.142) \text{\quad or \quad } (-0.142)-(-1)}
{{exercise_number}}. How much greater is the largest number than the smallest? The arrows mean \latex{ +(–7.2) }.
\latex{+5.48}
{{exercise_number}}. Replace the letters with numbers to make the equations true.
- \latex{(-2.8) + a = +0.5}
- \latex{(-13.05)- b = -20}
- \latex{(+7.93)-(+13.498) = c}
- \latex{d-(-37.2) = -9.87}
- \latex{(-29.5-4.3)-e =-40}
- \latex{f-(2.74-5.69) = +10}
{{exercise_number}}. Simplify the terms, then perform the additions.
- \latex{(-54.7) + (-25.3) -(-0.25)}
- \latex{(+7.42)- (-2.6) + (-3.456)}
- \latex{(-2)-(-5.28) + (+34.072)}
- \latex{(+47.9)-(+5.9)- (-52.1)}
{{exercise_number}}. The sum of \latex{ –4.9 } and \latex{ +16.3 } equals that of \latex{ –25.8 } and another number.
What is the other number?
{{exercise_number}}. Using numbers \latex{ –3.5 }, \latex{ –8.3 } and \latex{ +5.1 }, write down all the possible additions and subtractions using two of them in each case. Write down the greatest and smallest sum and difference. What is their sum?
{{exercise_number}}. Which decimal did I think of if
- it was subtracted from the sum of \latex{ 13.74 } and \latex{ 3.021 }, and I got \latex{ 8.307 };
- it was subtracted from \latex{ 7.029 }, then I subtracted \latex{ 7.029 } from the result, and I got \latex{ –13.8 ?}
{{exercise_number}}. Determine the rule and write down the previous three and the next three terms of the sequence.
... \latex{\fcolorbox{f6b900}{fef6e3}{$0.19$};\space} \latex{\fcolorbox{f6b900}{fef6e3}{$0.64$};\space} \latex{\fcolorbox{f6b900}{fef6e3}{$1.09$};\space ...}
Quiz
Calculate the following sums. (The terms are recurring decimals.)
- \latex{{0.\dot{3} +0.\dot{6} }}
- \latex{{0.\dot{6} +0.\dot{6} }}